One may see Lemma REF  (cf. Theorem 3.1 on p. 9 of [1]}) as an amalgam of two core insights. The first insight is that the square of the magnitude of the short-time Fourier transform of a bandlimited function is bandlimited itself. The second insight is that certain translates of Gaussians are complete in \(L^2([a,b])\) , when \(a < b\) . We note that the first insight allows for the application of the WSK sampling theorem in the time axis of the time-frequency plane while the second insight takes care of the sampling in the frequency axis. It is therefore interesting to think of the proof of Lemma REF  as a two step approach. First, time is discretised. Secondly, frequency is discretised.
